Variational Method for Vibration Analysis of Elliptic Cylinders Reinforced with Functionally Graded Carbon Nanotubes.

This study introduces a novel analytical framework for investigating the vibration characteristics of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) elliptical cylindrical shells under arbitrary boundary conditions. Unlike previous studies that focused on simplified geometries or specific boundary conditions, this work combines the least-squares weighted residual method (LSWRM) with an adapted variational principle, addressing high-order vibration errors and ensuring continuity across structural segments. The material properties are modeled using an extended rule of mixtures, capturing the effects of carbon nanotube volume fractions and distribution types on structural dynamics. Additionally, virtual boundary techniques are employed to generalize elastic boundary conditions, enabling the analysis of complex boundary-constrained structures. Numerical validation against existing methods confirms the high accuracy of the proposed framework. Furthermore, the influence of geometric parameters, material characteristics, and boundary stiffness on vibration behavior is comprehensively explored, offering a robust and versatile tool for designing advanced FG-CNTRC structures. This innovative approach provides significant insights into the optimization of nanoscale reinforced composites, making it a valuable reference for engineers and researchers in aerospace, marine, and construction industries.